Exercises for Section 10.1.  Basic Properties of Conformal Mappings

Exercise 1.  State where the following mappings are conformal.

1. (a) .
Solution 1 (a).

1. (b) .
Solution 1 (b).

1. (c) .
Solution 1 (c).

1. (d) .
Solution 1 (d).

1. (e) .
Solution 1 (e).

1. (f) .
Solution 1 (f).

Instructions For Exercises 2-5.
Find the angle of rotation and the scale factor of the mapping at the indicated points.

Exercise 2. at the points .
Solution 2.

Exercise 3. where at the points .

Remark. .
Solution 3.

Exercise 4.  ,  where ,  at the points .

Remark. .
Solution 4.

Exercise 5. at the points .
Solution 5.

Exercise 6.  Consider the mapping .

If ,  show that the lines are mapped onto orthogonal parabolas.
Solution 6.

Exercise 7.  Consider the mapping ,  where denotes the principal branch of the square root function.

If ,  show that the lines are mapped onto orthogonal curves.
Solution 7.

Exercise 8.  Consider the mapping .

Show that the lines are mapped onto orthogonal curves.
Solution 8.

Exercise 9.  Consider the mapping .

Show that the line segment ,  and the vertical line , where are mapped onto orthogonal curves.
Solution 9.

Exercise 10.  Consider the mapping ,  where denotes the principal branch of the logarithm function.

Show that the positive -axis and the vertical line are mapped onto orthogonal curves.
Solution 10.

Exercise 11.  (Indirectly Conformal Mapping)  Let be analytic at and .
Show that the function preserves the magnitude, but reverses the sense, of angles at .

Solution 11.

Exercise 12.  If is a mapping, where is not analytic, then what behavior would one expect regarding the angles between curves?
Solution 12.

(c) 2008 John H. Mathews, Russell W. Howell